| Preface | 5 |
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| Contents | 9 |
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| Part I Basic Stochastic Optimization Methods | 15 |
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| 1 Decision/Control Under Stochastic Uncertainty | 16 |
| 1.1 Introduction | 16 |
| 1.2 Deterministic Substitute Problems: Basic Formulation | 18 |
| 2 Deterministic Substitute Problems in Optimal Decision Under Stochastic Uncertainty | 22 |
| 2.1 Optimum Design Problems with Random Parameters | 22 |
| 2.2 Basic Properties of Substitute Problems | 31 |
| 2.3 Approximations of Deterministic Substitute Problems in Optimal Design | 33 |
| 2.4 Applications to Problems in Quality Engineering | 41 |
| 2.5 Approximation of Probabilities - Probability Inequalities | 42 |
| 2.6 Construction of State Functions in Structural Analysis and Design | 51 |
| Part II Differentiation Methods | 57 |
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| 3 Differentiation Methods for Probability and Risk Functions | 58 |
| 3.1 Introduction | 58 |
| 3.2 Transformation Method: Differentiation by Using an Integral Transformation | 61 |
| 3.3 The Differentiation of Structural Reliabilities | 69 |
| 3.4 Extensions | 71 |
| 3.5 Computation of Probabilities and its Derivatives by Asymptotic Expansions of Integral of Laplace Type | 76 |
| 3.6 Integral Representations of the Probability Function P(x) and its Derivatives | 86 |
| 3.7 Orthogonal Function Series Expansions I: Expansions in Hermite Functions, Case m = 1 | 89 |
| 3.8 Orthogonal Function Series | 103 |
| 3.9 Orthogonal Function Series Expansions III: Expansions in Trigonometric, Legendre and Laguerre Series | 105 |
| Part III Deterministic Descent Directions | 108 |
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| 4 Deterministic Descent Directions and Efficient Points | 110 |
| 4.1 Convex Approximation | 110 |
| 4.2 Computation of Descent Directions in Case of Normal Distributions | 116 |
| 4.3 Efficient Solutions ( Points) | 128 |
| 4.4 Descent Directions in Case of Elliptically Contoured Distributions | 132 |
| 4.5 Construction of Descent Directions by Using Quadratic Approximations of the Loss Function | 135 |
| Part IV Semi-Stochastic Approximation Methods | 141 |
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| 5 RSM-Based Stochastic Gradient Procedures | 142 |
| 5.1 Introduction | 142 |
| 5.2 Gradient Estimation Using the Response Surface Methodology (RSM) | 144 |
| 5.3 Estimation of the Mean Square (Mean Functional) Error | 155 |
| 5.4 Convergence Behavior of Hybrid Stochastic Approximation Methods | 160 |
| 5.5 Convergence Rates of Hybrid Stochastic Approximation Procedures | 166 |
| 6 Stochastic Approximation Methods with Changing Error Variances | 190 |
| 6.1 Introduction | 190 |
| 6.2 Solution of Optimality Conditions | 191 |
| 6.3 General Assumptions and Notations | 192 |
| 6.4 Preliminary Results | 196 |
| 6.5 General Convergence Results | 203 |
| 6.6 Realisation of Search Directions | 217 |
| 6.7 Realization of Adaptive Step Sizes | 232 |
| 6.8 A Special Class of Adaptive Scalar Step Sizes | 249 |
| Part V Technical Applications | 264 |
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| 7 Approximation of the Probability of Failure/Survival in Plastic Structural Analysis and Optimal Plastic Design | 266 |
| 7.1 Introduction | 266 |
| 7.2 Probability of Survival/Failure p8,Pf | 267 |
| 7.3 Approximation of ps,Pf by Linearization of the Transformed Limit State Function | 270 |
| 7.4 Computation of the ß-Point z | 275 |
| 7.5 Trusses | 278 |
| 7.6 Reliability- Based Design Optimization ( RBDO) | 282 |
| Part VI Appendix | 286 |
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| A Sequences, Series and Products | 288 |
| A.1 Mean Value Theorems for Deterministic Sequences | 288 |
| A.2 Iterative Solution of a Lyapunov Matrix Equation | 296 |
| B Convergence Theorems for Stochastic Sequences | 300 |
| B.1 A Convergence Result of Robbins-Siegmund | 300 |
| B.2 Convergence in the Mean | 303 |
| B.3 The Strong Law of Large Numbers for Dependent Matrix Sequences | 305 |
| B.4 A Central Limit Theorem for Dependent Vector Sequences | 306 |
| C Tools from Matrix Calculus | 308 |
| C.1 Miscellaneous | 308 |
| C. 2 The v. Mises- Procedure in Case of Errors | 309 |
| Index | 322 |
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